Understanding the Dirac Equation and the ... - Progress in Physics

Volume 4

PROGRESS IN PHYSICS

October, 2013

Understanding the Dirac Equation and the Electron-Vacuum System William C. Daywitt National Institute for Standards and Technology (retired), Boulder, Colorado. E-mail: [email protected]

It has been close to a century since the Dirac equation first appeared, but it has yet to be understood on an intuitive, fundamental level. The reason for this lack of understanding is twofold: the equation is expressed in terms of the secondary constant ℏ; and the vacuum state and its coupling to the electron particle have not been developed as part of the electron model. What follows briefly reviews the vacuum coupling and illustrates it by deriving the Schrödinger and Pauli equations as derivatives of the Dirac equation, and by explaining the zitterbewegung response that is a vacuum dynamic associated with the coupling force. It is argued that the fields of quantum electrodynamics have as their origin the degenerate vacuum state.

operates. Thus the spinors are associated with PV distortion — with no distortion the gradients vanish, resulting in null The Dirac electron defined here is a massive “point” charge spinors and the dissolution of (3). (−e∗ , m) that obeys the Dirac equation and is coupled to the The free-particle Dirac equation can be expressed in the negative-energy Planck vacuum (PV) continuum via the twoform (from (A10) in Appendix A) term coupling force [1] ( ) (→ ) ( ) − σ · irc ∇χ ∂ ϕ ϕ e2∗ mc2 irc (3) + → = − − (1) c∂t χ −χ σ · ir ∇ϕ c 2 r r 1 Introduction

the massive charge exerts on the PV. The electron Compton radius rc (= e2∗ /mc2 ) is that radius from the center of the massive charge (in its rest frame) to the radius rc where the coupling force vanishes. The bare charge (−e∗ ) itself is massless, while the electron mass m results from the bare charge being driven by the zero-point electromagnetic field [2] [3]; corresponding to which is a vanishingly small sphere containing the driven charge whose center defines the center of both the driven charge and its derived mass. It is from the center of this small sphere that the position operator r for the massive charge and the electron-vacuum complex is defined and from which the radius r in (1) emerges. The PV model of the complete electron consists of two interdependent dynamics, the dynamics of the massive charge in the previous paragraph and the dynamics of the PV continuum to which the massive charge is coupled. An example of the latter dynamic is the (properly interpreted) zitterbewegung [4] [1] that represents a harmonic-oscillator-type excitation taking place at the r = rc sphere surrounding the massive point charge, an oscillation resulting from the vacuum response to the vanishing of (1) at rc . The point-like nature of the massive charge, in conjunction with the continuum nature of the PV, are what give the electron its so-called waveparticle-duality. Mathematically, the electron’s wave nature is apparent from the fact that the spinor solutions to the Dirac equation are spinor fields, and it is upon these fields that the covariant gradient operator

in terms of the single constant rc , a constant that normalizes the operator in (2). The free-space particle solution ϕ, and the negative-energy vacuum solution χ, for this electron-vacuum − system are 2×1 spinors and → σ is the Pauli 2×2 vector matrix. The spinor solutions from the two simultaneous equations in (3) are strongly coupled by the inverted χ-ϕ spinor configuration of the second term, showing the vacuum state to be an integral part of the electron phenomenon. (It will be seen that this coupling is even present in the nonrelativistic Schrödinger equation.) The negative spinor (−χ) on the right is a manifestation of the negative-energy nature of the vacuum. Equation (3) expresses the Dirac equation in terms of the normalized PV gradients on the left of the equal sign. What follows illustrates the previous ideas by reiterating the standard development of the free-particle Schrödinger equation and the minimal coupling substitution leading to the Pauli equation.

( ) ∂ ∂ ,∇ ∂µ = µ = ∂x c∂t

) ( ) ( ) (→ − σ · irc ∇χ0 0 ∂ ϕ0 = + → irc − −2χ0 c∂t χ0 σ · irc ∇ϕ0

78

(2)

2 Schrödinger equation The Dirac-to-Schrödinger reduction [5, p. 79] begins with eliminating the high-frequency components from (3) by assuming ( ) ( ) ( ) ϕ ϕ0 −imc2 t/ℏ ϕ0 −ict/rc = e = e (4) χ χ0 χ0 where ϕ0 and χ0 are slowly varying functions of time compared to the exponentials. Inserting (4) into (3) gives (5)

William C. Daywitt. Understanding the Dirac Equation and the Electron-Vacuum System

October, 2013

PROGRESS IN PHYSICS

where the 0 on the right is a 2×1 null spinor. This zero spinor indicates that the mass energy of the free particle is being ignored, while the effective negative-mass energy of the “vacuum particle” has been doubled. In effect, mass energy for the particle-vacuum system has been conserved by shifting the mass energy of the free particle to the vacuum particle. The lower of the two simultaneous equations in (5) can be reduced from three to two terms by the assumption ir ∂χ0 ≪ |−2χ | (6) 0 c c∂t

Volume 4

3 Minimal coupling By itself the coupling force (1) is insufficient to split the twofold degeneracy of the spinors in the free-particle Dirac (3) and Schrödinger (12) equations. It takes an external field to effect the split and create the well-known 1/2-spin electron states. The following illustrates this conclusion for the case of the minimal coupling substitution. The minimal coupling substitution [5, p.78] is pµ −→ pµ − eAµ /c

(14)

where e is the magnitude of the observed electron charge, if the kinetic energy (from the first equation in (A2)) of the pµ = (E/c, p) is the 4-momentum, and Aµ = (A0 , A) is the vacuum particle is significantly less than its effective mass electromagnetic 4-potential. Inserting (14) with (A1) and energy. Inserting (6) into (5) yields (A2) into the Dirac equation (A3) leads to ( ) (→ ) ( ) − ( ) ( ) σ · irc ∇χ0 ∂ ϕ0 0 eA ∂ irc (7) + → = − α· b p− iℏ − eA0 ψ − cα ψ = mc2 βψ (15) c∂t 0 −2χ0 σ · irc ∇ϕ0 ∂t c as the nonrelativistic version of (3). The mass energy of the which can be expressed as free particle, and the kinetic energy of the vacuum particle ( ) (→ ) ( ) ( ) − σ · (irc ∇ + a)χ ∂ ϕ ϕ ϕ (associated with the lower-left null spinor), are discarded in irc (16) + → = a0 + − the Schrödinger approximation. c∂t χ χ −χ σ · (irc ∇ + a)ϕ Separating the two equations in (7) produces in the 2×1 spinor formulation, where a0 ≡ eA0 /mc2 and a ≡ ∂ϕ0 → eA/mc2 . Then proceeding as in Section 2 produces irc +− σ · irc ∇χ0 = 0 (8) c∂t ( ) (→ ) ( ) ( ) − σ · (irc ∇ + a)χ0 ∂ ϕ0 ϕ0 0 irc + → = a0 + . (17) and − c∂t 0 χ0 −2χ0 σ · (irc ∇ + a)ϕ0 → − (9) σ · irc ∇ϕ0 = −2χ0 The Compton radius in (16) and (17) has been accounted and inserting (9) into (8) leads to for as a gradient normalizer. The remaining constants (e and m) appear only in association with the 4-potential Aµ — if − ∂ϕ0 (→ σ · irc ∇)2 irc − ϕ0 = 0 . (10) the external potential vanishes, the electron charge and mass c∂t 2 are removed (a0 = 0 and a = 0) from the equations, and (16) and (17) reduce to (3) and (7) respectively. Furthermore, Finally, inserting the Pauli-matrix identity (A12) the energy eA0 appears to increase the energy level of the − (→ σ · irc ∇)2 = I (irc ∇)2 (11) negative-energy PV continuum. This latter conclusion can be appreciated by combining the two terms on the right side into (10) yields the free-particle Schrödinger equation of (17): ) ( ( ) eA0 /mc2 ϕ0 2 2 ∂ϕ0 (irc ∇) ∂ϕ0 (iℏ∇) (18) ( ) irc (12) = ϕ0 or iℏ = ϕ0 eA0 /mc2 − 2 χ0 c∂t 2 ∂t 2m where the two spin components in ϕ0 are ignored in this ap- where a0 has been replaced 2by its definition. With a constant potential energy eA0 = 2mc , the lower parenthesis vanishes proximation. The scalar harmonic function and the free-space electron energy and the vacuum-energy ϕ0 −→ exp [−i(Et − p · r)/ℏ)] (13) spectrum just begin to overlap [1]. This latter result is the phenomenon that leads to the relativistic Klein paradox [5, satisfies both equations as it should, and leads to the nonrel- p. 127]. If it is further assumed that ativistic energy-momentum relation E = p2 /2m, where p = mv. The equation on the left in (12) expresses the Schrödinger (19) |a0 χ0 | ≪ |− 2χ0 | equation in terms of PV gradients. The vacuum property implied by (11), and the fact that ϕ0 then (17) becomes is a spinor field, show that the vacuum state is a significant ) ( ) ( ) ( ) (→ − (but hidden) part of the nonrelativistic Schrödinger equation. σ · (irc ∇ + a)χ0 ϕ0 0 ∂ ϕ0 = a0 + + → irc (20) − The Dirac-to-Pauli reduction leads to the same conclusion. 0 −2χ0 c∂t 0 σ · (irc ∇ + a)ϕ0 William C. Daywitt. Understanding the Dirac Equation and the Electron-Vacuum System

79

Volume 4

PROGRESS IN PHYSICS

October, 2013

which is the nonrelativistic version of (16). Then eliminating Appendix A: Dirac equation χ0 from the two simultaneous equations in (20) leads to the The PV is characterized in part by the two Compton relaequation tions [1] rc mc2 = r∗ m∗ c2 = e2∗ (= cℏ) (A1) → − → − σ · (irc ∇ + a) σ · (irc ∇ + a) ∂ϕ0 irc = ϕ0 + a0 ϕ0 (21) connecting the massive point charge (−e∗ , m) of the electron c∂t 2 to the individual Planck particles (−e∗ , m∗ ) within the degenfor the spinor ϕ0 . Equation (21) then leads to the Pauli equaerate PV, where rc and m, and r∗ and m∗ are the Compton tion [5, p.81]. radius and mass of the electron and Planck particles respecUsing (A11) to calculate the square of the numerator in tively. The bare charge (−e∗ ) is massless and is related to the the first term on the right of the equal sign in (21) yields observed electronic charge (−e) via the fine structure constant α = e2 /e2∗ . From (A1), the energy and momentum operators − i→ σ · (irc ∇ × a) ∂ϕ0 (irc ∇ + a)2 = ϕ0 + ϕ0 + a0 ϕ0 (22) can be expressed as irc c∂t 2 2 ( ) ∂ ∂ 2 remembering that ϕ0 post-multiplies the square before calcub and b p = −iℏ∇ = mc(−irc ∇) (A2) E = iℏ = mc irc ∂t c∂t lation. The first term in (22) contains the electron’s orbital angular momentum; and the second its spin, as manifested in the parenthetical factors implying that the operators, operat− the scaler product of → σ and the curl of the vector potential A. ing on the Dirac spinors, provide a measure of the gradients Using (A1), the corresponding spin operator can be expressed within the PV continuum. In the present free-electron case, as these gradients are caused solely by the coupling force (1) − − − σ e2→ ℏ→ σ (−e∗ )(−e∗ )→ σ b s= (23) and its negative (Appendix B). = ∗ = 2 2c 2c The upper and lower limits to the PV negative-energy where one of the charges (−e∗ ) in (23) belongs to the massive spectrum are −mc2 and −m c2 respectively, where m is the ∗ ∗ point charge (−e∗ , m) and the other to the separate Planck par- Planck mass. The continuum nature of the vacuum is an apticles (−e∗ , m∗ ) within the PV. The product e2∗ suggests that proximation that applies down to length intervals as small as the spin may be related to the interaction of the massive point ten Planck lengths (10 r ) or so; that is, as small as ∼ 10−32 cm. ∗ charge with the PV charges when the vacuum is under the Using (A1) and (A2), the Dirac equation [5, p.74] influence of a magnetic field B (= ∇ × A). ∂ψ ie2∗ + α · ie2∗ ∇ ψ = mc2 βψ (A3) 4 Conclusions and comments c∂t The physics of the PV state [1, 6] has provided a simple intu- can be expressed as itive explanation for the Dirac, Schrödinger, and Pauli equa∂ψ tions in terms of the massive point charge (−e∗ , m) and its + α · irc ∇ ψ = βψ (A4) irc c∂t interaction (1) with the PV. It is the ignorance of this coupling force that has obscured the meaning of the Dirac equa- where the 4×4 vector-matrix operator tion since its inception and, as seen in the next paragraph, the ( ) − 0 → σ meaning of the zitterbewegung frequency. α= → (A5) − 2 σ 0 The electron Compton relation rc m = e∗ in (A1) holds for both combinations (∓e∗ , ±m); so the vacuum hole (e∗ , −m) − where → σ = (σ1 , σ2 , σ3 ) and exerts a coupling force on the vacuum state that is the negative ( ( ( ) ) ) of (1). The combination of the two forces explain why the 0 1 0 −i 1 0 σ1 = σ2 = σ3 = (A6) zitterbewegung frequency (2c/rc [1] [4]) is twice the angular 1 0 i 0 0 −1 2 frequency (mc /ℏ = c/rc ) associated with the electron mass are the three 2×2 Pauli matrices. The 4×4 matrix operator (from Appendix B). ( ) The purpose of this paper is to illustrate the massiveI 0 β = (A7) charge-PV nature of the electron phenomenon; and to reestab0 −I lish the vacuum state as an essential and necessary part of a complete electron theory, that part that has been superseded where I represents the 2×2 unit matrix and the zeros here and by the idea of the quantum field. While the quantum field for- in (A5) are 2×2 null matrices. The covariant gradient operator ( ) ( ) malism, like the Green function formalism, is an important ∂ ∂ ∂ ∂ ∂ ∂ = = ,∇ (A8) , , , tool [5, p. 143] [7], the present author believes that the corre∂xµ c∂t ∂x0 ∂x1 ∂x2 ∂x3 sponding quantum field does not constitute an essential physical phenomenon apart from the dynamics of vacuum state is seen in (A2) and (A4) to have its differential coordinates normalized (∂xµ /rc ) by the electron Compton radius. (from Appendix C). 80


October, 2013

PROGRESS IN PHYSICS

Volume 4

The 4×1 spinor wavefunction ψ can be expressed for small ∆r/rc . This yields the harmonic oscillator motion as [5, p. 79] from Newton’s second law ( ) ϕ ( )2 ( 2 ) ψ= (A9) e∗ c d2 ∆r χ ∆r = − =− ∆r (B2) rc dt2 mrc3 where ϕ and χ are the usual 2×1 spinors, and where the two components in each represent two possible spin states. The with the “spring constant” (e2 /r3 ) and oscillator frequency ∗ c spinor ϕ is the free-space particle solution and χ is the neg- c/rc . The corresponding motion that is due to the vacuum ative-energy hole solution. Inserting (A9) into (A4), and car- hole (e∗ , −m) (whose charge and mass fields exert a force that rying out the indicated matrix operations, yields the Dirac is the negative of (1)) is equation ( ) (→ ) ( ) ( )2 − σ · irc ∇χ ∂ ϕ ϕ c d2 ∆r irc + → = (A10) ∆r (B3) − 2 =+ − c∂t χ −χ σ · irc ∇ϕ rc dt in terms of the 2×1 spinors. showing that the massive free charge and the vacuum hole The following is an important property of the Pauli matricause identical accelerations within the PV continuum. → − ces, and the PV state (because of σ): the vector Pauli matrix The total vacuum acceleration is the sum of (B2) and (B3) → − σ obeys the identity [5, p.12] ( 2 ) ( )2 e∗ d2 ∆r c → − → − → − ( σ · a)( σ · b) = I a · b + i σ · a × b (A11) = −2 ∆r = −2 ∆r (B4) 2 3 rc dt mrc − where a and b both commute with → σ, but are otherwise arwith the corresponding harmonic oscillator frequency bitrary three-vectors. Using (A11) (with a = b = rc ∇) leads √ to √ c 2 2 2e2∗ → − ( σ · rc ∇) = I (rc ∇) (A12) = 2 (B5) 3 rc mrc which connects the normalized ∇ operator in the relativistic Dirac equation to the same operator in the nonrelativistic which is √2 times the angular frequency associated with the Schrödinger equation. electron mass energy. Given the roughness of the calculaInserting the operators from (A2) into (A10) and rear- tions, this result implies that the combined massive-charge ranging the result leads to the two simultaneous equations forces, acting simultaneously on the PV continuum, are the ( ) source of the zitterbewegung with its 2c/rc frequency. 2 → − b − mc ϕ = c σ · b E pχ (A13) Appendix C: Quantum field and ( ) 2 − b + mc χ = c→ E σ ·b pϕ. (A14) The PV is envisioned as a degenerate negative-energy sea of fermionic Planck particles. Because of this degeneracy, the b + mc2 ) and using (A14) vacuum experiences only small displacements from equilibThen, pre-multiplying (A13) by (E and (A11) leads to rium when stressed. Thus the displacements due to the coupling force (1) are small, and so the potential energy corre( ) b2 − m2 c4 ϕ = c2 b E p2 ϕ (A15) sponding to the stress can be approximated as a quadratic in those displacements. This important result enables the vacand, after reversing the process, to an identical equation for χ. uum to support normal mode coordinates and their assumed Thus both ϕ and χ separately obey the Klein-Gordon equation quantum fields, as explained in the simple demonstration to [5, p.31]. follow. The normal mode connection [8, pp. 109–119] to the Appendix B: Zitterbewegung frequency quantum field can be easily understood by examining a string, The following rough heuristic argument identifies the two stretched between two fixed points in a stationary reference coupling forces that explain why the zitterbewegung frequen- frame, that exhibits small transverse displacements from cy [1, 4] is twice the angular frequency (mc2 /ℏ = c/rc ) asso- equilibrium. In this case, the corresponding potential energy ciated with the electron mass energy. can be expressed in terms of quadratic displacements. If the The force the massive point charge (−e∗ , m) exerts on the displacements are represented by the function ϕ(t, x) at time t PV is given by equation (1) which, using r = rc + ∆r and and position x along the string, then the quadratic assumption rc = e2∗ /mc2 , leads to implies that the displacements must obey the wave equation ( 2) 2 3 2 2 (e /r )∆r e∗ e mc 1 ∂2 ϕ ∂2 ϕ =− ∗ c − ≈ − 3∗ ∆r (B1) = 2 (C1) 2 2 rc + ∆r (rc + ∆r) (1 + ∆r/rc ) rc c2 ∂t2 ∂x William C. Daywitt. Understanding the Dirac Equation and the Electron-Vacuum System

81

Volume 4

PROGRESS IN PHYSICS

October, 2013

where c is a propagation velocity. The string geometry leads At this point the quantum-field formalism discards the to the Fourier series representation preceding foundation upon which the fields are derived, and assumes that the fields themselves are the primary reality N ∑ [8, p. 119]. Part of the reason for this assumption is that, in ϕ(t, x) = an (t) sin (nπx/L) (C2) the past, no obvious foundation was available. However, the n=1 demonstration here provides such a foundation on the simple, for the standing wave on the string, where L is the string but far-reaching assumption that the vacuum is a degenerate state which can sustain a large stress without a correspondlength. Inserting (C2) into (C1) produces ingly large strain. a¨ n (t) = −ω2n an (t) where ωn = nπc/L (C3) Submitted on September 11, 2013 / Accepted on September 21, 2013

and where the amplitude an (t) is that of a harmonic oscillator. The constant characterizing the Dirac equation is the Compton radius rc . So it is reasonable to set the string length L ∼ rc to determine the fundamental frequency ω1 = πc/L in (C3). Furthermore, the harmonics of ω1 can have wavelengths of the order of the Planck length r∗ (antiparticle excitation is, of course, ignored in this rough argument); so the length L can be subdivided

References 1. Daywitt W.C. The Electron-Vacuum Coupling Force in the Dirac Electron Theory and its Relation to the Zitterbewegung. Progress in Physics, 2013, v. 3, 25. 2. Puthoff H.E. Gravity as a Zero-Point-Fluctuation Force. Physical Review A, 1989, v. 39, no. 5, 2333–2342. 3. Daywitt W.C. The Source of the Quantum Vacuum. Progress in Physics, 2009, v. 1, 27. 4. Barut A.O. and Bracken A.J. Zitterbewegung and the Internal Geometry of the Electron. Physical Review D, 1981, v. 23, no. 10, 2454–2463.

L rc N= ∼ minimum length division of string r∗ 3.86 × 10−11 ∼ 1022 (C4) 1.62 × 10−33 times, and ϕ in (C2) can be expressed as an integral if convenient since r∗ ≪ rc . The total energy of the vibrating string can thus be expressed as ( )2  ∫ L  ( )2 ρ 2 ∂ϕ   ρ ∂ϕ (C5) E= + c   dx 2 ∂t 2 ∂x  0 =

5. Gingrich D.M. Practical Quantum Electrodynamics. CRC, The Taylor & Francis Group, Boca Raton, London, New York, 2006. 6. Daywitt W.C. The Planck Vacuum. Progress in Physics, 2009, v. 1, 20. See also www.planckvacuum.com. 7. Milonni P.W. The Quantum Vacuum—an Introduction to Quantum Electrodynamics. Academic Press, New York, 1994. 8. Aitchison I.J.R., Hey A.J.G. Gauge Theories in Particle Physics Vol. 1. Taylor & Francis, New York, London, 2003.

which, inserting (C2) into (C5), results in [8, p.117] E=

] N [ L ∑ ρ˙a2n ρω2n a2n + 2 n=1 2 2

(C6)

where the first and second terms in (C5) and (C6) are the kinetic and potential string energies respectively (ρ is the string density). The crucial significance of (C6) is that it is a sum of independent normal-mode energies, where the an (t) are the normal mode coordinates. From this normal mode setting, the quantum field energy E=

N ( ∑ n=1

) ) N ( ∑ 1 1 2 nn + ℏωn = mc rc k n nn + 2 2 n=1

(C7)

is defined, where nn is the number of normal modes associated with the wavenumber kn = ωn /c. In effect, the integers nn (⩾ 0) determine the quantized energy level of each normal mode oscillator an (t). The 1/2 component in (C7) is the zero-point energy of the string-vacuum system. 82
